Differentiating With respect To x

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SUMMARY

This discussion clarifies the concept of differentiation in calculus, specifically focusing on differentiating functions with respect to x. When differentiating an explicit function where y is a function of x, the derivative dy/dx represents the rate of change of y relative to x. The derivative dx/dx equals 1, indicating that x changes at the same rate as itself. This understanding confirms that the derivative measures how one variable changes in relation to another.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically differentiation.
  • Familiarity with explicit functions where y is defined as a function of x.
  • Knowledge of the notation used in calculus, such as dy/dx and dx/dx.
  • Basic comprehension of rates of change and their significance in calculus.
NEXT STEPS
  • Study the rules of differentiation, including the product and quotient rules.
  • Learn about implicit differentiation and its applications.
  • Explore the concept of higher-order derivatives and their interpretations.
  • Investigate real-world applications of derivatives in physics and engineering.
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Students studying calculus, educators teaching mathematical concepts, and anyone seeking to deepen their understanding of differentiation and its applications in various fields.

Bashyboy
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So, when you differentiate a function, specifically an explicit function, where y is a function of x, you are differentiating each term with respect with x. Well, when you differentiate x with respect with x, does that mean you are trying to find out how x changes with x? What does that mean anyways? And when you differentiate y with respect to x, does that mean you are trying to figure out how y is changing with each value of x?
 
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I am confused as to what you are looking for. However dx/dx = 1.
 
Yes, when you find "the derivative of y with respect to x" you are comparing the (instantaneous) rate of change of y compared to the instantaneous rate of change of x or "how fast y changes as x changes". The "derivative of x with respect to x", then, compares how fast x changes to how fast x changes and, because obviously, they change at the same rate, that derivative is, as mathman says, 1.
 
Oh, brilliant. Thank you both. That has confirmed my thoughts.
 

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